System for optimization of golf clubs

ABSTRACT

Methods of optimizing and matching golf clubs, and the clubs so matched. Clubs are selected to produce a desired relationship between hand speed at impact (tempo), and centripetal force on the hands at impact (perceived force). Clubs may also be selected to minimize and equate (or reduce variation in) the radius of gyration, as measured about a center point which is individually determined for each golfer.

FIELD OF THE INVENTION

[0001] The present invention relates to a system for constructing a setof matched golf clubs which are optimized for a particular golfer.

BACKGROUND OF THE INVENTION

[0002] The rules of golf allow a golfer to carry a golfer to carry amaximum of fourteen clubs: an assortment of “woods” and “irons,” andusually one “putter.” Golfers select clubs to produce shots ofpredetermined lengths (carry plus roll) and trajectories. Once clubshave been selected, the golfer's goal is to swing each club in a mannerthat gives predictable results. It is recognized that the golfer's jobis made easier if the same (or a similar) swing works for all clubs: a“repeatable” swing. It is further recognized that a repeatable swing isfacilitated by having all of the clubs “feel the same” (or similar)during the swing. A set of clubs which feel the same to the golfer issaid to be “matched.”

[0003] Club manufacturers offer a vast array of clubs for golfers tochoose from. Within a set, manufacturers typically seek to match clubs(often separating woods and irons) by having the club heads have similarshape and be made of similar materials. Additionally, sets usually haveuniform grips and shafts (material and flexibility) and somemanufacturers allow the golfer to customize club length and/or lie.

[0004] In addition to matching these basic parameters, many attemptshave been made to match clubs in other ways. “Swing weighting” systemsgenerally match clubs according to their first inertial moment about anarbitrarily selected point twelve inches (“Lorythmic scale”) or fourteeninches (“Official scale”) from the butt end of the club (see, U.S. Pat.No. 4,128,242 to Elkins, Jr. and U.S. Pat. No. 4,887,815 to Hughes etal.). Related systems have been proposed to balance the club about apoint closer to the grip, e.g., five inches from the butt end of theshaft (see, U.S. Pat. No. 4,674,324 to Benoit). Other methods includematching the first inertial moment and/or the second inertial moment(see, U.S. Pat. No. 4,415,156 to Jorgensen), the radius of gyration(see, U.S. Pat. No. 4,674,324 to Benoit and U.S. Pat. No. 5,277,059 toChastonay), or the vibrational frequencies (see, U.S. Pat. No. 4,555,112to Masghati and U.S. Pat. No. 5,722,899 to Cheng) over the set of clubs.

[0005] With the exception of one system that seeks to equalize thedynamic moment of inertia of the club including the moment of inertia ofthe golfer's arms (see, U.S. Pat. No. 5,769,733 to Williams, et al.),all currently known club matching systems focus on matching clubs to aselected mathematical standard with no attempt to incorporate thephysical and swing characteristics of a particular golfer. Since golfersdiffer in many ways not addressed by mathematical standards, it is notsurprising that it remains an elusive goal for a golfer to obtain a setof clubs which truly provide him with a uniform swing feel. An improvedmethod of selecting and matching clubs to optimize a golfer's game isneeded.

SUMMARY OF THE INVENTION

[0006] The present invention provides a method of designing a matchedset of golf clubs which takes into account both objective parameterswhich control the motion of the ball in response to an impact from aclub head, and subjective parameters related to individual preferencesin “feel” and idiosyncracies of a particular golfer's swing.

[0007] In one aspect, the invention comprises a method of designing agolf club, comprising detemining relationships relating tempo (how fastthe golfer swings) and perceived force (how much force is applied alongthe shaft) to club length and club head mass for a particular golfer.Two parameters are selected from the following group of three: targetdistance, club length (and shaft flexibility), and preferred trajectoryfor the ball, and then these parameters are used with the tempo andperceived force functions to determine the third parameter and the clubhead mass for the golf club design. Tempo may be measured by the speedof the golfer's hands at the moment of impact, and perceived force bythe centripetal force along the shaft of the club at the moment ofimpact. In order to determine these values, an effective arm length maybe determined, for example by measuring the actual arm length of thegolfer, or the distance from the golfer's hands to his sternum orcollarbone in address position. Trajectory may be controlled by varyingthe club head loft, which may be the design loft (i.e., manufacturer'sintended loft) or the effective loft (i.e., impact loft). Exemplaryfunctions for tempo and perceived force with club length (or club lengthplus arm length) include constant, linear, or power-law relationships.The method may comprise designing multiple clubs fitting the tempo andperceived force functions. One or more of the clubs may be weighted inorder to reduce the variation in perceived length over the multipleclubs. Perceived length may be determined by the radius of gyrationaround a selected center point, which may be determined as describedbelow. A club design method according to this aspect of the inventionmay include designing a set of up to thirteen golf clubs, all of whichobey selected tempo and perceived force functions and having reducedvariation in perceived length over the set. The method may furtherinclude constructing the clubs, using, for example, a CAD/CAM system.

[0008] In another aspect, the invention includes a method of determiningthe personal center of gyration of a golfer, by having the golfer swinga test club and a weighted comparison club. First the golfer swings thetest club to feel its perceived length. Then the golfer swings acomparison club one or more times, while weight is added to thecomparison club at a selected point on the shaft, until the clubs feelas though they have the same club length to the golfer. The point aroundwhich the test club and the weighted comparison club have the sameradius of gyration is then determined. This aspect of the inventionfurther includes constructing a plurality of clubs having reducedvariation in their radii of gyration around the center point determinedaccording to this method.

[0009] In a further aspect, the invention includes a matched set ofclubs (or a subset) for a golfer. Each club of the matched set has atarget distance (or preferred trajectory) and a club length, and thetempo of the clubs when swung by the golfer to achieve the targetdistances has a selected functional relationship to the club length (therelationship should actually constrain the club design). The functionalrelationship may be, for example, constant, linear, polynomial, orpower-law. In a related aspect, the invention comprises a matched set ofgolf clubs which have a selected relationship between perceived forceand club length when swung to achieve a set of target distances (orpreferred trajectories). Perceived force may be determined by thecentripetal force applied along the shaft at impact, and the tempo maybe determined by the speed of the golfer's hands at impact. Preferably,clubs have selected functional relationships for both tempo andperceived force. Clubs may further have optimized lean angle asdescribed in U.S. patent application Ser. No. 08/248,515. Clubs may beconstructed, for example, using a CAD/CAM system.

[0010] As it is used herein, the term “club length” means the length ofa club as measured from the butt end of the shaft to the center of massof the club head.

DETAILED DESCRIPTION OF THE DRAWING

[0011] Preferred embodiments of the invention are described withreference to the single figure of the drawing, which

[0012]FIG. 1 plots of club head mass as a function of club length forthree sets of clubs according to the invention; and

[0013]FIG. 2 is a schematic view of a club weighted to adjust the radiusof gyration according to the invention.

DETAILED DESCRIPTION

[0014] A logical starting point (although not the only possible startingpoint) in the design of a set of clubs is for the golfer to specifytarget distances (carry and roll) and trajectories for up to 13 clubs(leaving one space in the set for a putter) that he could expect toachieve with reasonable consistency with a full swing given his physicalparameters and swing characteristics. Thus, for example, a reasonablyadvanced golfer might specify target distances of 250, 225, 200, 187.5,175, 162.5, 150, 137.5, 125, 112.5, 100, 87.5, and 75 yards,respectively. He could also specify desired trajectories. In thepreferred embodiments of the invention, desired trajectory is fullydescribed either by the design loft of the club head (the manufacturer'sintended loft), or by the effective loft of the club head (the actualloft at impact) each as more fully defined in copending and commonlyassigned U.S. patent application Ser. No. 09/248,515, filed Feb. 8,1999, which is incorporated herein by reference.

[0015] For a particular type of golf ball, each target distance can beachieved by an infinite number of combinations of club head masses andlofts, club lengths and shaft flexibilities, and swing speeds. A table(hereinafter, the Table) could be constructed that, to any degree offineness desired, shows how varying these parameters varies the distance(and trajectory) achieved. An automatic ball striking machine could beused to construct such a Table. Alternatively, the Table could beconstructed by carefully observing the results of golfers hitting ballswith various clubs. In yet another embodiment, the values in the Tablemight be determined by analytical calculations.

[0016] A golfer may wish to base his target distances and trajectoriesoff the longest distance he can consistently achieve (typically with adriver). Conventional wisdom suggests that driving distance isrelatively insensitive to club head weight for most golfers (see, forexample, Cochran and Stobbs, Search for the Perfect Swing, TriumphBooks, Chicago, 1968). Therefore, club head weight can be held at asuitable constant level (e.g., 200 g) while optimizing club length (andshaft flex) and club head loft. Alternatively, the player may be allowedto vary club head weight in the following procedure in order todetermine a maximum driving distance.

[0017] Using drivers of varying length (and possibly also of varyingshaft flexibility), the length which allows the golfer to generate themaximum club head speed (where, ceteris paribus, speed is a surrogatefor impact force) can be determined using various known devices for themeasurement of club head speed at impact. Once the maximum consistentlyachievable club head speed is known, the other parameters (e.g., loft,club head weight, etc.) can be varied (using analytical or empiricalmethods, or by having a golfer attempt to swing the club at a constantspeed) to determine the combination of parameters that achieves thegolfer's maximum driving distance. Knowing his maximum driving distancewould allow a golfer to choose useful target distances for the remainingclubs.

[0018] The present invention suggests that two parameters should be usedto select amongst possible clubs for each desired target distance andtrajectory: the force the golfer wants to feel at the time he strikesthe ball with the club (his “perceived force” function), and how fast hewants to swing the club (his “tempo” function). The invention furtherteaches that a golfer will be better able to achieve a repeatable swing(i.e., generate the desired perceived force at the desired tempo overand over again) if the variation in how long different clubs feel duringthe swing (the “perceived length” function) is reduced relative to priorart clubs. In the preferred embodiments of the invention, perceivedforce is measured by the centripetal force at the point of impact of theclub head with the ball, and tempo is measured by the speed of thegolfer's hands (also at the point of impact). In other preferredembodiments of the invention, perceived length is measured by the radiusof gyration calculated around a golfer's personal center (describedbelow).

[0019] In one preferred embodiment of the invention, tempo and perceivedforce are maintained as constants over the set of clubs. This embodimenthas been found to lead to club heads that are lighter than those usualin the art. In another preferred embodiment, tempo increases linearlywith club length plus arm length over the set of clubs, while perceivedforce remains constant. The motivation for this preferred embodimentcomes from the fact that many golfers swing their longer clubs (forlonger shots) with more abandon (i.e., faster) than their shorter clubs,and thus prefer to feel a faster hand speed for the longer clubs, whileretaining the same feeling of force at impact. This embodiment has beenfound to lead to club heads that are heavier than those usual in theart. An embodiment wherein tempo is constant and centripetal forceincreases linearly with length produces constant weight club heads; anembodiment wherein tempo is constant and centripetal force increasesfaster than linearly produces club heads that increase in weight as theclub length increases. Finally, an embodiment wherein tempo increaseswith the square root of club length plus arm length, while perceivedforce remains constant, maintains the ability of the golfer to swinglonger clubs with greater abandon, and produces club heads with weightssimilar to those of prior art clubs.

[0020] The first two preferred embodiments described in the paragraphabove can be considered to be zero-parameter and one-parameter models ofa golfer's preferred swing characteristics. Higher-order models can alsobe used, e.g., polynomial or power-law models. The model chosen willdepend upon the preferences of the individual golfer, and may bedetermined subjectively or analytically, for example, by video analysisof the golfer's swing to determine a preferred relationship between handspeed, centripetal force, club length, and club head mass. The number ofparameters should be small enough to actually constrain the clubselection.

[0021] Regardless of the swing technique used by the golfer, at impactthe left (leading) arm and the shaft of the club typically will bevirtually in line and moving together like an extended shaft. This factallows determination of the speed of the golfer's hand at impact (thetempo) if his “effective” arm length is known. (The effective arm lengthmay be approximated by the actual length of the golfer's arms, or by alength measured from the top of the sternum to the hands when the clubis at address position, or by other appropriate measurements of thegolfer's body). That is, knowing the impact speed of two club heads(e.g., the driver and another club), the lengths of the clubs and thegolfer's effective arm length allows the calculation of hand speed andcentripetal force for each club at impact, using the followingequations: $\begin{matrix}{{v_{hand}({driver})} = {v_{driverhead}\frac{l_{arm}}{l_{driver} + l_{arm}}}} & (1) \\{{v_{hand}({club})} = {v_{clubhead}\frac{l_{arm}}{l_{club} + l_{arm}}}} & (2) \\{{F_{cent}({driver})} = {m_{driverhead}\frac{v_{driverhead}^{2}}{l_{driver} + l_{arm}}}} & (3) \\{{F_{cent}({club})} = {m_{clubhead}\frac{v_{clubhead}^{2}}{l_{club} + l_{arm}}}} & (4)\end{matrix}$

[0022] where v_(hand)(driver) and v_(hand)(club) represent the speed ofthe hands when swinging the driver and the second club, respectively,l_(arm), l_(driver), and l_(club) club represent the golfer's effectivearm length and the lengths of the driver and the second club,respectively, V_(driverhead) and v_(clubhead) represent the velocitiesat impact of the driver head and the second club head, respectively, andm_(driverhead) and m_(clubhead) represent the masses of the driver headand the second club head, respectively. In this embodiment, centripetalforce is calculated based only on the contribution due to the mass ofthe club head (justification for this assumption can be found, forexample, in Cochran and Stobbs, Search for the Perfect Swing, p.145-147, which suggests that at impact the club head can be modeled asbeing suspended from a length of string). More rigorous computations ofcentripetal force will be readily apparent to those skilled in the art.

[0023] Using the model to determine r_(tempo) and r_(force) (the ratiosof hand speed and centripetal force of the two clubs), and solving forthe mass of the club head (for a specified club length), we calculate$\begin{matrix}{{r_{tempo} \equiv \frac{v_{hand}({club})}{v_{hand}({driver})}} = {\frac{v_{clubhead}}{v_{driverhead}}\frac{l_{driver} + l_{arm}}{l_{club} + l_{arm}}}} & (5) \\{{r_{force} \equiv \frac{F_{cent}({club})}{F_{cent}({driver})}} = {\frac{m_{clubhead}}{m_{driverhead}}\left( \frac{v_{clubhead}}{v_{driverhead}} \right)^{2}\frac{l_{driver} + l_{arm}}{l_{club} + l_{arm}}}} & (6) \\{m_{clubhead} = {m_{driverhead}\frac{r_{force}}{r_{tempo}^{2}}\left( \frac{l_{driver} + l_{arm}}{l_{club} + l_{arm}} \right)}} & (7)\end{matrix}$

[0024] A number of possible models for the variation of r_(tempo) andr_(force) with club length and/or club mass are contemplated. Exemplarymodels include linear relationships (e.g., r_(tempo)=l_(club)/l_(driver)or r_(tempo)=(l_(club)+l_(arm))/(l_(driver)+l_(arm))), polynomialrelationships (e.g., r_(tempo) ^(∝)(l_(club)/l_(driver))²+c₀(l_(club)/l_(driver))), and power-lawrelationships (e.g., r_(tempo)=(l_(club)/l_(driver))^(k) orr_(tempo)=(l_(club)+l_(arm)/l_(driver +l) _(arm))^(k)). In particular, asquare root relationship in whichr_(tempo)=((l_(club)/+l_(arm))/(l_(driver)+l_(arm)))^(1/2) andr_(force)=1 has been found to give excellent results for the inventor.FIG. 1 shows relationships between club head mass and club length forthe cases of r_(tempo)=r_(force)=1 (20), r_(tempo)=(l_(club)+l_(arm)/(l)_(driver)+l_(arm)) and r_(force)=1 (22), andr_(tempo)=((_(club)+l_(arm))/(l_(driver+l) _(arm)))^(1/2) andr_(force)=1 (24). (In this Figure, points represent individual clubs,and the line represents the ideal functional relationship, assuming anarm length of 33″ for the golfer).

[0025] For example, suppose a golfer with a 33″ arm length chooses a 48″driver and a 214 gram club head. If he chooses a 36″ wedge, and desiresconstant centripetal force, the club head weights he would use in theconstant, linearly increasing, and square root relationships betweenhand speed and club length plus arm length would be 251, 346 and 295grams, respectively. The golfer would complete the wedge design by goingto the Table at the target distance and choosing the loft that, whencombined with the calculated club head mass, club length (and shaftflexibility), and club head speed, produces the target distance for thewedge. Alternatively, the golfer could specify the desired loft (i.e.,trajectory) of the wedge instead of its club length. The Table couldonce again be used to solve for the missing parameter that satisfies thetarget distance while maintaining the desired tempo and perceived force.

[0026] It will be understood by those skilled in the art that theselected relationships between tempo, perceived force, and club lengthmay not be constant over an entire set of clubs. For example, differentrelationships may obtain over the irons and the woods, or over othersubsets of a full set of clubs, without departing from the principles ofthe invention.

[0027] Further optimization may also be applied to some or all of a setof clubs whether or not they have been optimized as described above. Forexample, the lean angles of the clubs may be adjusted to equate theeffective lofts and the design lofts, as described in copending andcommonly assigned U.S. patent application Ser. No. 09/248,515, filedFeb. 8, 1999, which is incorporated herein by reference. In addition,further optimization may be achieved by reducing the variation of theperceived length of the set or subsets of clubs, as described below.

[0028] In one preferred embodiment, perceived length is approximated bythe radius of gyration (ROG). There are many plausible centers aroundwhich ROG can be measured for the purpose of matching a set of clubs,ranging from the shoulder to the elbow to the wrist to points on theshaft of the club. Previous patents (e.g., U.S. Pat. No. 5,277,059 toChastonay) have used particular centers for these calculations, but ineach case the center point was assumed to be the same for all golfers.The golf swing is a complex muscular action with many moving parts thatmust be carefully coordinated and sequenced. It is therefore notsurprising that prior art, with its arbitrarily selected center of ROG,has not produced a matching system in which “matched” clubs feelsubstantially the same for all golfers. The present inventionencompasses the realization that the center of the ROG may be a functionof the swing style of a particular golfer, and that a personalizedcenter can be measured for each golfer and used for further optimizationand club matching. That is, it is believed that it is beneficial todynamically match a set of clubs by reducing the variation in orequating their ROGs around the golfer's personal center, and furtherthat it is beneficial for the ROG to be quite short, as compared to thatof conventional clubs. These aims can be accomplished in an embodimentof the invention as follows.

[0029] First, a golfer swings his shortest club (e.g., a wedge), andmentally notes its feel. He then swings a relatively long club (e.g., adriver), and weight (e.g., lead tape) is added at a specified pointalong the shaft of the driver (usually below where the hands grip theclub, relatively close to where the grip ends) until the two clubs feelas though they are the same length when swung. (The two clubs used inthis test will preferably but not necessarily have been optimized fortempo and perceived force as described above). Using the magnitude ofthe added weight, the center of the ROG can be determined analytically,by constructing an equation for the ROG for each club about an unknowncenter point X. The ROG of the two clubs can then be set equal to solvefor X, the golfer's personal center. All remaining clubs in the set canthen be weighted so as to equate the ROG to that of the two measuredclubs. Alternatively, clubs may be weighted to equate ROG only acrosssubsets of the clubs. A club 10 weighted to adjust the ROG is shown inFIG. 2. The weight 12 is placed on the shaft 14 of the club 10. Theorigin at the butt end of the shaft O, arbitrary position X(positive, asshown), the weight 12 position p_(weight), and the length of the clubl_(club) are all shown.

[0030] It should be noted that golfers are often not used to thinking interms of perceived length, and steps should be taken to explain theperceived length concept so that the golfer can accurately identifyclubs having the same perceived length. It may be useful to have thegolfer swing prior art (i.e., unweighted) clubs, for example a wedge anda driver, and to identify the difference in feel as a difference inperceived length. Additionally, the golfer could use a “baseball batswing” test, in which the golfer (optionally blindfolded) swings clubshorizontally (to avoid hitting the ground for long clubs whichnevertheless “feel” short). The baseball bat swing test may either beused directly to test clubs, or may be used to help the golfer identifythe “length” perception for clubs which are subsequently swung in a moretraditional manner. Alternatively, the golfer can be asked to swingclubs whose ROG variation has been reduced for typical center locations,so that the difference in feel between such modified clubs andtraditional clubs can be identified.

[0031] For example, suppose the ROG optimization procedure is followedwith the 36″ wedge and the 48″ driver described above, assuming a linearrelationship for tempo. Weight is added to the driver at a point, forexample, 12″ from the butt end of the shaft, until the clubs areperceived as having the same length. If the added weight has a mass of178 g, the radius of gyration about a point X inches from the end of theshaft is $\begin{matrix}{{{ROG}({driver})} = \sqrt{\frac{{m_{driverhead}\left( {l_{driver} - X} \right)}^{2} + {m_{driverweight}\left( {p_{weight} - X} \right)}^{2}}{m_{driverhead} + m_{driverweight}}}} & (8)\end{matrix}$

[0032] where p_(weight) is the position of the added weight, in inchesfrom the butt end of the shaft, and m_(driverweight) is the mass of theadded weight. (Those skilled in the art will note that the contributionsof the shaft and grip to the ROG have been neglected in the aboveformulation, as have variations due to the fact that the club head andthe added weight are not actually point masses. It will be apparent thatthe above formulae can easily be modified to take these contributionsinto account). Thus, for the driver described, the ROG in terms of X is{square root}{square root over (((214 g)(48−X)²+(178 g)(12−X)²)/(214g+178 g)})={square root}{square root over (X²−63.3X+1320)}. In thissimple model, the ROG of the wedge is simply the distance from X to thehead of the wedge, in this case, 36″−X. Solving for X, we find thatX=−3. (That is, the golfer's personal center is found at a point on hisarms 3″ above the butt end of the club), and the ROG of both clubs is39″. These parameters may then be used to weight all of the other clubsto match the ROG of the clubs about the golfer's personal center. (Addedweights at the same point on the driver which would lead todeterminations of X=0, X=5, or X=10 would be 187 g, 208 g, and 245 g,respectively).

[0033] It is convenient but not necessary to start the process with theshortest club or to assume that the shortest club receives no weight. Inone embodiment an arbitrary club is chosen and weight added until thegolfer's swing speed begins to suffer. That is, weight is added whichreduces the ROG to the minimum acceptable value for that golfer and thatclub. Then another club is chosen and weight added to it until thegolfer feels that the second club has the same perceived length as thefirst. Once again the center point X can be analytically determined andthe ROG calculated and used to calibrate other clubs. It is possible,however, that the calculations may call for the removal of weight fromsome clubs (which is usually infeasible). Additionally, the weightsadded in those embodiments of the invention which minimize ROG can berather large, and may not be to the liking of all golfers, although itis believed that greater accuracy and consistency will be achievableonce golfers become accustomed to the clubs of the invention.

[0034] It will be apparent to those skilled in the art that the weightsneed not all be placed at the same location on each club shaft, nor isit necessary to place only a single weight on the shaft. Similarcalculations will apply, and the appropriate weight to be placed at anypoint or points on the shaft can be found for each club. If the golferwishes all his clubs to have the same weight, this can generally beaccommodated by solving for the position at which an appropriate weightshould be added. It may be, however, that this position will not be onthe shaft, in which case the design will have to be modified.

[0035] For example, ignoring possible weight differences of the gripsand shafts, if the wedge has a head weight of 370 g, and the driver hasa head weight of 214 g, a 156 g weight can be placed on the drivershaft, and slid along the shaft until the clubs feel equal. (In theabove case, mathematically this would occur when the weight is placed3.28″ from the butt end of the shaft). However, if the club head weightof the wedge is only 295 g, then there is no position where a 29 gweight can be placed on the driver to equate the ROGs. In this case, theROGs may be equated by weighting both clubs. For example, if a 125 gweight is placed on the wedge at a position 27″ from the butt end of theshaft (making the total weight 420 g), then a 206 g weight may be placed1.7″ from the butt end of the shaft of the driver to give both clubs anROG of 36.6 and a total weight of 420 g (still using the personal centerof X=−3).

[0036] In some cases, it may not be possible to equalize ROG over theentire set without the individual clubs becoming unacceptably heavy. Inthese cases, the ROG may be equalized for subsets of the clubs or may bemade to vary in a predictable way (e.g., linearly) over the set, so asto at least reduce the variation of the ROG over the set (or a subset)of clubs. “Reduced variation” in ROG, as that phrase is used herein,means a variation in ROG less than that of a set of clubs having thesame club head masses and club lengths, but having no additional weightsplaced on the shafts.

[0037] The measurement techniques described for measurement of the ROGmay also have utility in determining the “effective arm length” used inthe optimization of club lengths, head weights, and lofts. The perceivedcenter may or may not be the same for the ROG and for the centripetalforce, but similar techniques involving holding all parameters but oneconstant and finding the closest match to a known club may be used.

[0038] It will be seen that the reduction and matching process for ROGdoes not change the head weight or length of the club. It can thereforeeasily be combined with the optimization techniques for matching tempoand perceived force described above as well as adjustments to equatedesign and effective loft, as described in copending and commonlyassigned U.S. patent application Ser. No. 09/248,515.

[0039] Additional weights could be used to match the first and/or secondinertial moments, either about the same center as that measured for theROG or about a different center measured by similar methods.

[0040] The methods of the invention provide a great deal of data onoptimization of golf clubs for a particular user. Given the personalizednature of the data, it may be difficult or impossible to findcommercially available club heads in the style preferred by the golferwhich satisfy some or all of the criteria of the invention (possiblyincluding the criteria set forth in U.S. patent application Ser. No.09/248,515). It may therefore be desirable to make club heads fromscratch, and to incorporate these heads into golf clubs having desiredclub lengths, shaft flexibilities, grips, added weights, etc. Forexample, a club head can be modeled using a computer-aided design (CAD)system. The basic input to the system would be the golfer's preferredclub head design, which he could either specify or choose from optionsalready in the system. The appropriate lie, loft, head mass, lean angle(see U.S. patent application Ser. No. 09/248,515), and any otherparameters would then be determined, and the basic design modifiedaccordingly. (If necessary, heads may be hollow and/or made ofcomposites in order to tailor the mass). A computer-assistedmanufacturing (CAM) system can then be used to make the designed head,which may be made in multiple parts if necessary. Suitable CAM systemsare commercially available which use investment casting, sand casting,electrochemical machining, electrical discharge machining,computer-controlled conventional machining systems, or other methods offorming one-of-a-kind metal items. Depending on the accuracy of themethod used and the level of detail in the design, additionalpost-processing machining, etching, or polishing may be necessary. Thismethod allows construction of a truly unique set of clubs, optimized fora single golfer.

EXAMPLE

[0041] Table 1 shows the specification for a set of clubs optimized forthe inventor. The inventor is 5′8″ tall, and has an arm length of 33″.The location of his personal center for ROG calculations has beendetermined to be 7.567″ below the butt end of the shaft. His clubspecifications, all calculated according to the invention (i.e.,constant centripetal force and hand speed increasing with the squareroot of club length plus arm length), are as follows: TABLE 1 Weight atClub Clubhead 10″ from ROG Target Length Weight shaft end (inchesDistance (inches) (grams) Club Type (grams) from X) (yards) 48 214driver 220 28.44 250 46 225 3-wood 187 28.45 225 44 237 5-wood 153 28.44200 42 250 4-iron 117 28.45 187.5 41⅓ 254 5-iron 105 28.43 175 40⅔ 2596-iron 93 28.42 162.5 40 263 7-iron 80 28.43 150 39⅓ 268 8-iron 67 28.43137.5 38⅔ 273 9-iron 54 28.43 125 38 279 wedge 41 28.43 112.5 37⅓ 284pitching wedge 27 28.45 100 36⅔ 289 sand wedge 14 28.42 87.5 36 295 lobwedge 0 28.43 75

[0042]FIG. 3 plots the club lengths and club head masses specified inTable 1, and compares them to calculated values for a square rootrelationship in whichr_(tempo)=((l_(club)+l_(arm))/(l_(driver)+l_(arm)))^(1/2) andr_(force)=1. The shafts of these clubs are all regular flex (R) graphiteshafts.

[0043] The added weights at the point 10 inches from the butt end of theshaft in the clubs described above are simply lead tape wrapped aroundthe shaft and secured. This method of weighting the shaft has theadvantage that the added mass is easily tailored and applied, but someplayers may prefer that some or all of the weight be added in theinterior (rather than the exterior) of the shaft. It should be notedthat equating the ROG over an entire set of clubs may require ratherheavy weights on the longer clubs (as demonstrated by the data of Table1), which may not be practical to place solely in the shaft interior.Depending on the material used for the weights and the available spaceinside the shaft, embodiments of the invention placing the weights inthe interior of the shaft may only be able to equate the ROG oversubsets of the clubs.

[0044] These clubs differ from standard clubs in many ways, including

[0045] The tempo is matched (square root relationship);

[0046] The perceived force is matched (constant);

[0047] The perceived length is matched (constant).

[0048] Further, the clubs are significantly longer than standard clubs(especially the woods), and contain substantial added weight. Also, thisset of clubs intentionally does not include a 2-iron or 3-iron.

[0049] Other embodiments of the invention will be apparent to thoseskilled in the art from a consideration of the specification or practiceof the invention disclosed herein. It is intended that the specificationand examples be considered as exemplary only, with the true scope andspirit of the invention being indicated by the following claims.

What is claimed is:
 1. A method of designing a customized golf club,comprising: determining a tempo function relating tempo to club lengthfor a particular golfer; determining a perceived force function relatingperceived force to club length and club head mass for the golfer;selecting two design parameters from the group consisting of targetdistance for the club; club length and shaft flexibility for the club;and preferred trajectory for a golf ball; and using the selected designparameters, together with the determined tempo and perceived forcefunctions, to calculate optimum values for the unselected designparameter and the club head mass for the customized golf club.
 2. Themethod of claim 1, wherein tempo is measured by speed of the golfer'shands at impact.
 3. The method of claim 1, wherein perceived force ismeasured by centripetal force applied along the shaft at impact.
 4. Themethod of claim 3, further comprising measuring an effective arm lengthof the golfer in order to determine the centripetal force.
 5. The methodof claim 4, wherein the effective arm length is selected from the groupconsisting of arm length, distance from hands to sternum in addressposition, and distance from hands to collarbone in address position. 6.The method of claim 1, wherein trajectory is controlled by varying clubhead loft.
 7. The method of claim 6, wherein the club head loft is thedesign loft.
 8. The method of claim 6, wherein the club head loft is theeffective loft.
 9. The method of claim 1, further comprising optimizingthe lean angle of the clubs.
 10. The method of claim 1, wherein thetempo is independent of club length.
 11. The method of claim 1, whereinthe tempo is a linear function of club length or of club length plus armlength.
 12. The method of claim 1, wherein the tempo is a power-lawfunction of club length or of club length plus arm length.
 13. Themethod of claim 1, wherein the perceived force is independent of clublength.
 14. The method of claim 1, wherein the perceived force is alinear function of club length or of club length plus arm length. 15.The method of claim 1, wherein the perceived force is a power-lawfunction of club length or of club length plus arm length.
 16. Themethod of claim 1, wherein the perceived force is independent of clublength and the tempo is proportional to the square root of club lengthplus arm length.
 17. The method of claim 1, further comprising designinga second golf club having at least one different design parameter fromthe first club, wherein the same tempo function and perceived forcefunction apply to both clubs.
 18. The method of claim 17, wherein thetwo golf clubs have a reduced difference in perceived length.
 19. Themethod of claim 18, wherein the perceived length is measured bydetermining the radius of gyration of a club about a selected centerpoint.
 20. The method of claim 19, wherein the center point is selectedby having the golfer swing a test club to determine its perceivedlength; having the golfer swing a comparison club one or more timeswhile adding weight to the comparison club at a selected point along theshaft until the golfer is unable to distinguish the perceived lengths ofthe test club and the comparison club; and determining the center pointaround which the test club and the weighted comparison club haveidentical radii of gyration.
 21. The method of claim 1, comprisingdesigning up to thirteen golf clubs all having the same tempo andperceived force functions and reduced differences in perceived lengths.22. The method of claim 1, further comprising constructing the designedclub using a CAD/CAM system.
 23. A method of determining a perceivedcenter of gyration for a golfer, comprising having the golfer swing atest club to determine its perceived length; having the golfer swing acomparison club one or more times while adding weight to the comparisonclub at a selected point along the shaft until the golfer is unable todistinguish the perceived lengths of the test club and the comparisonclub; and determining the center point around which the test club andthe weighted comparison club have identical radii of gyration.
 24. Amethod of constructing a matched set of golf clubs for a golfer,comprising determining a perceived center of gyration according to themethod of claim 23; and constructing a plurality of clubs having reducedvariation in radius of gyration with respect to the determined centerpoint.
 25. A matched set of golf clubs for a golfer, comprisingplurality of golf clubs, wherein each club has a length and a targetdistance, wherein a selected functional relationship exists over theplurality of clubs between the tempo when striking a ball with one ofthe clubs to produce the target distance for that club, and the lengthof that club.
 26. The matched set of golf clubs of claim 25, wherein thetempo when striking a ball with each of the clubs to produce that club'starget distance is the same.
 27. The matched set of golf clubs of claim25, wherein the selected functional relationship is a linearrelationship.
 28. The matched set of golf clubs of claim 25, wherein theselected functional relationship is a polynomial relationship.
 29. Thematched set of golf clubs of claim 25, wherein the selected functionalrelationship is a power-law relationship.
 30. The matched set of golfclubs of claim 25, wherein a selected functional relationship existsover the plurality of clubs between the perceived force when striking aball with one of the clubs to produce the target distance for that club,and the length of that club.
 31. The matched set of golf clubs of claim30, wherein the perceived force is measured by determining thecentripetal force exerted along the shaft of the club at impact.
 32. Thematched set of golf clubs of claim 25, wherein the tempo is measured bydetermining the speed of the golfer's hands at impact.
 33. The matchedset of golf clubs of claim 25, wherein the lean angle of the clubs isoptimized.
 34. The matched set of golf clubs of claim 25, wherein theclubs have been designed by a computer-aided design method, and whereinthe club have been manufactured by a computer-assisted manufacturingmethod.
 35. A matched set of golf clubs for a golfer, comprising aplurality of golf clubs, wherein each club has a length and a targetdistance, wherein a selected functional relationship exists over theplurality of clubs between the perceived force when striking a ball withone of the clubs to produce the target distance for that club, and thelength of that club.
 36. A matched set of golf clubs for a golfer,comprising a plurality of golf clubs, wherein each club has a length anda desired ball trajectory, wherein a selected functional relationshipexists over the plurality of clubs between the tempo when striking aball with one of the clubs to produce the desired ball trajectory forthat club, and the length of that club.
 37. The matched set of golfclubs of claim 36, wherein the tempo when striking a ball with each ofthe clubs to produce that club's desired ball trajectory is the same.38. The matched set of golf clubs of claim 36, wherein the selectedfunctional relationship is a linear relationship.
 39. The matched set ofgolf clubs of claim 36, wherein the selected functional relationship isa polynomial relationship.
 40. The matched set of golf clubs of claim36, wherein the selected functional relationship is a power-lawrelationship.
 41. The matched set of golf clubs of claim 36, wherein aselected functional relationship exists over the plurality of clubsbetween the perceived force when striking a ball with one of the clubsto produce the desired ball trajectory for that club, and the length ofthat club.
 42. The matched set of golf clubs of claim 41, wherein theperceived force is measured by determining the centripetal force appliedalong the shaft of the club at impact.
 43. The matched set of golf clubsof claim 36, wherein the tempo is measured by determining the speed ofthe golfer's hands at impact.
 44. The matched set of golf clubs of claim36, wherein the lean angle of the clubs is optimized.
 45. A matched setof golf clubs for a golfer, comprising a plurality of golf clubs,wherein each club has a length and a desired ball trajectory, wherein aselected functional relationship exists over the plurality of clubsbetween the perceived force when striking a ball with one of the clubsto produce the desired ball trajectory for that club, and the length ofthat club.